We explain how Itô Stochastic Differential Equations on manifolds may be defined as 2-jets of curves and show how this relationship can be interpreted in terms of a convergent numerical scheme. We use jets as a natural language to express geometric properties of SDEs. We explain that the mainstream choice of Fisk-Stratonovich-McShane calculus for stochastic differential geometry is not necessary. We give a new geometric interpretation of the Itô-Stratonovich transformation in terms of the 2-jets of curves induced by consecutive vector flows. We discuss the forward Kolmogorov equation and the backward diffusion operator in geometric terms. In the one-dimensional case we consider percentiles of the solutions of the SDE and their properties. In particular the median of a SDE solution is associated to the drift of the SDE in Stratonovich form for small times.

The standard model of information geometry, expressed as Fisher-Rao metric and Amari-Chensov tensor, reflects an embedding of probability density by log-transform. The standard embedding was generalized by one-parametric families of embedding function, such as α-embedding, q-embedding, κ-embedding. Further generalizations using arbitrary monotone functions (or positive functions as derivatives) include the deformed-log embedding (Naudts), U-embedding (Eguchi), and rho-tau dual embedding (Zhang). Here we demonstrate that the divergence function under the rho-tau dual embedding degenerates, upon taking ρ = id, to that under either deformed-log embedding or U-embedding; hence the latter two give an identical divergence function. While the rho-tau embedding gives rise to the most general form of cross-entropy with two free functions, its entropy reduces to that of deformed entropy of Naudts with only one free function. Fixing the gauge freedom in rho-tau embedding through normalization of dual-entropy function renders rho-tau cross-entropy to degenerate to U cross-entropy of Eguchi, which has the simpler property, not true for general rho-tau cross-entropy, of reducing to the deformed entropy upon setting the two pdfs to be equal. In Part I, we investigate monotone embedding in divergence function, entropy and cross-entropy, whereas in the sequel (Part II), in induced geometries and probability families.

It is well-known that a contrast function defined on a product manifold M x M induces a Riemannian metric and a pair of dual torsionfree affine connections on the manifold M. This geometrical structure is called a statistical manifold and plays a central role in information geometry. Recently, the notion of pre-contrast function has been introduced and shown to induce a similar differential geometrical structure on M, but one of the two dual affine connections is not necessarily torsion-free. This structure is called a statistical manifold admitting torsion. This paper summarizes such previous results including the fact that an estimating function on a parametric statistical model naturally defines a pre-contrast function to induce a statistical manifold admitting torsion and provides some new insights on this geometrical structure. That is, we show that the canonical pre-contrast function can be defined on a partially flat space, which is a flat manifold with respect to only one of the dual connections, and discuss a generalized projection theorem in terms of the canonical pre-contrast function.

This paper is an overview of results that have been obtain in [2] on the convex regularization of Wasserstein barycenters for random measures supported on Rd. We discuss the existence and uniqueness of such barycenters for a large class of regularizing functions. A stability result of regularized barycenters in terms of Bregman distance associated to the convex regularization term is also given. Additionally we discuss the convergence of the regularized empirical barycenter of a set of n iid random probability measures towards its population counterpart in the real line case, and we discuss its rate of convergence. This approach is shown to be appropriate for the statistical analysis of discrete or absolutely continuous random measures. In this setting, we propose an efficient minimization algorithm based on accelerated gradient descent for the computation of regularized Wasserstein barycenters.

We study some extensions of the empirical likelihood method, when the Kullback distance is replaced by some general convex divergence or j-discrepancy. We show that this generalized empirical likelihood method is asymptotically valid for general Hadamard differentiable functionals.

Stochastic Development Regression using Method of Moments Line Kühnel, Stefan Sommer GSI2017

This paper considers the estimation problem arising when inferring parameters in the stochastic development regression model for manifold valued non-linear data. Stochastic development regression captures the relation between manifold-valued response and Euclidean covariate variables using the stochastic development construction. It is thereby able to incorporate several covariate variables and random effects.
The model is intrinsically de ned using the connection of the manifold, and the use of stochastic development avoids linearizing the geometry.
We propose to infer parameters using the Method of Moments procedure that matches known constraints on moments of the observations conditional on the latent variables. The performance of the model is investigated in a simulation example using data on nite dimensional landmark manifolds.

There are two geometrical structures in a manifold of probability distributions. One is invariant, based on the Fisher information, and the other is based on the Wasserstein distance of optimal transportation. We propose a uni ed framework which connects the Wasserstein distance and the Kullback-Leibler (KL) divergence to give a new information-geometrical theory. We consider the discrete case consisting of n elements and study the geometry of the probability simplex Sn-1, the set of all probability distributions over n atoms. The Wasserstein distance is introduced in Sn-1 by the optimal transportation of commodities from distribution p ∈ Sn-1 to q ∈ Sn-1. We relax the optimal transportation by using entropy, introduced by M. Cuturi (2013) and show that the entropy-relaxed transportation plan naturally de nes the exponential family and the dually at structure of information geometry.

Although the optimal cost does not de ne a distance function, we introduce a novel divergence function in Sn-1, which connects the relaxed Wasserstein distance to the KL-divergence by one parameter.

Stochastic Development Regression using Method of Moments (slides)GSI2017

This paper is concerned with the computation of an optimal matching between two manifold-valued curves. Curves are seen as elements of an infinite-dimensional manifold and compared using a Riemannian metric that is invariant under the action of the reparameterization group. This group induces a quotient structure classically interpreted as the "shape space". We introduce a simple algorithm allowing to compute geodesics of the quotient shape space using a canonical decomposition of a path in the associated principal bundle. We consider the particular case of elastic metrics and show simulations for open curves in the plane, the hyperbolic plane and the sphere.

We study a data-driven sub-Riemannian (SR) curve optimization model for connecting local orientations in orientation lifts of images. Our model lives on the projective line bundle R2 x P1, with P1 = S1/~ with identification of antipodal points. It extends previous cortical models for contour perception on R2 x P1 to the data-driven case. We provide a complete (mainly numerical) analysis of the dynamics of the 1st Maxwell-set with growing radii of SR-spheres, revealing the cutlocus. Furthermore, a comparison of the cusp-surface in R2 x P1 to its counterpart in R2 x S1 of a previous model, reveals a general and strong reduction of cusps in spatial projections of geodesics. Numerical solutions of the model are obtained by a single wavefront propagation method relying on a simple extension of existing anisotropic fast-marching or iterative morphological scale space methods. Experiments show that the projective line bundle structure greatly reduces the presence of cusps. Another advantage of including R2 x P1 instead of R2 x S1 in the wavefront propagation is reduction of computational time.

This paper addresses the drone tracking problem, using a model based on the Frenet-Serret frame. A kinematic model in 2D, representing intrinsic coordinates of the drone is used. The tracking problem is tackled using two recent filtering methods. On the one hand, the Invariant Extended Kalman Filter (IEKF), introduced in [1] is tested, and on the other hand, the second step of the filtering algorithm, i.e. the update step of the IEKF is replaced by the update step of the Unscented Kalman Filter (UKF), introduced in [2]. These two filters are compared to the well known Extended Kalman Filter. The estimation precision of all three algorithms are computed on a real drone tracking problem.

For data with non-Euclidean geometric structure, hypothesis testing is challenging because most statistical tools, for example principal component analysis (PCA), are specific for linear data with a Euclidean structure. In the last 15 years, the subject has advanced through the emerging development of central limit theorems, first for generalizations of means, then also for geodesics and more generally for lower dimensional subspaces. Notably, there are data spaces, even geometrically very benign, such as the torus, where this approach is statistically not feasible, unless the geometry is changed, to that of a sphere, say. This geometry is statistically so benign that nestedness of Euclidean PCA, which is usually not given for the above general approaches, is also naturally given through principal nested great spheres (PNGS) and even more flexible than Euclidean PCA through principal nested (small) spheres (PNS). In this contribution we illustrate applications of bootstrap two-sample tests for the torus and its higher dimensional generalizations, polyspheres.

This paper introduces a novel algorithm for the online estimate of the Riemannian mixture model parameters. This new approach counts on Riemannian geometry concepts to extend the well-known Titterington approach for the online estimate of mixture model parameters in the Euclidean case to the Riemannian manifolds. Here, Riemannian mixtures in the Riemannian manifold of Symmetric Positive De nite (SPD) matrices are analyzed in details, even if the method is well suited for other manifolds.

Symplectic structure is powerful especially when it is applied to Hamiltonian systems.We show here how this symplectic structure may de ne and evaluate an integer index that measures the defect for the system to be Hamiltonian. This defect is called the Geometric Degree of Non Conservativeness of the system. Darboux theorem on di erential forms is the key result. Linear and non linear frameworks are investigated.

During wakefulness and deep sleep brain states, cortical neural networks show a di erent behavior, with the second characterized by transients of high network activity. To investigate their impact on neuronal behavior, we apply a pairwise Ising model analysis by inferring the maximum entropy model that reproduces single and pairwise moments of the neuron's spiking activity. In this work we rst review the inference algorithm introduced in Ferrari, Phys. Rev. E (2016) [1]. We then succeed in applying the algorithm to infer the model from a large ensemble of neurons recorded by multi-electrode array in human temporal cortex. We compare the Ising model performance in capturing the statistical properties of the network activity during wakefulness and deep sleep. For the latter, the pairwise model misses relevant transients of high network activity, suggesting that additional constraints are necessary to accurately model the data.